System for wide bandwidth damping

ABSTRACT

A positive electronic feedback circuit for mechanical vibration damping. The feedback circuit is connected from an output transducer fixed to a vibratable structure to an input transducer fixed to the structure. Preferably the feedback circuit has an output signal with a phase about 90 degrees lagging its input signal. Three conditions set forth herein make this possible.

BACKGROUND OF THE INVENTION

This invention relates to vibration analysis, and more particularly toan active damping system for vibratable mechanical structures.

PRIOR ART STATEMENT

Narrow bandwidth damping is known in the prior art. For example, see thefollowing printed publications:

"Theoretical and Experimental Results on Active Vibration Dampers" by T.H. Rockwell and J. M. Lawther (Journal of the Acoustical Society ofAmerica, Vol. 36, No. 8, August, 1964).

"Electronic Damping of Orthogonal Bending Modes in a Cylindrical Mast"by R. L. Forward (American Institute of Aeronautics and Astronautics,Inc., AIAA 81-4018, Vol. 18, No. 1, Jan.-Feb., 1981, pp. 5-17).

The Shock and Vibration Bulletin No. 53, Part 4 (The Shock and VibrationInformation Center of the Naval Research Laboratory, May, 1983).

SUMMARY OF THE INVENTION

In accordance with the system of the present invention, there isprovided an electronic feedback circuit which produces a feedback signalwhich lags the phase of the feedback circuit input signal by betweenabout 0 and 180 degrees and therefore provides positive feedback in eachof the individual resonant modes of vibration of a structure for whichvibration damping is sought. Preferably, the feedback signal lags thephase of the circuit input signal by 90 degrees.

Damping is accomplished by satisfying three conditions described indetail hereinafter.

BRIEF DESCRIPTION OF THE DRAWINGS

In the drawings which are to be regarded as merely illustrative:

FIG. 1 is a diagrammatic view of the system of the present invention;

FIG. 2 is a schematic diagram of one specific embodiment of the presentinvention;

FIG. 3 is a graph illustrating a curve fitting and step required toverify circuit values for the invention;

FIG. 4 is a schematic diagram of a second specific embodiment of thepresent invention;

FIG. 5 is a graph of a variable proportional to the imaginary part ofone Laplace transfer function versus a function proportional tofrequency;

FIG. 6 is a graph of an imaginary part of the product of two Laplacetransfer functions versus a function proportional to frequency;

FIG. 7 is a graph of a variable proportional to the imaginary part ofstill another Laplace transfer function versus a function proportionalto frequency; and

FIG. 8 is a graph of an imaginary part of the product of still two otherLaplace transfer functions versus a function proportional to frequency.

DESCRIPTION OF THE PREFERRED EMBODIMENT

In the drawings, in FIG. 1, a source 10 supplies an input voltage V_(in)(t) where t is time.

A summing circuit 11, a mechanical structure 12 to be damped, an inputpiezoelectric transducer 13, an output piezoelectric transducer 14, andan electronic feedback circuit 15 are also provided.

Structure 12 may be a window or an aluminum bulkhead or a wall. For theexamples given herein, it may be assumed that structure 12 is analuminum block.

Summing circuit 11 impresses an input voltage upon input transducer 13proportional to the sum of the output voltages of source 10 and feedbackcircuit 15.

Transducers 13 and 14 may be bonded or otherwise fixed to structure 12in any conventional way.

Output transducer 14 is connected to an output junction 16, the samebeing connected to the input of feedback circuit 15. As will beexplained, a Laplace transfer function G(ω) is obtained from a curvefitting process performed after measuring, from time to time, thevoltage at V_(out) (t) appearing at junction 16.

The Laplace transfer function H(ω) of electronic feedback circuit 15 maybe defined as: ##EQU1##

This equation sets forth H(ω) with transducers 13 and 14 to dampmultiple mode structure 12. One of the structural modes has an undampedresonant frequency of ω₁ and θ₁ percent of critical damping. The closedloop circuit has input voltage V_(in) (t) and voltage V_(out) (t). Thepoles P_(n) and zeroes Z_(m) of feedback circuit 15 can be real orcomplex, Re(P_(n))≧0.

One embodiment of the invention is shown in FIG. 2 with a source 17, asumming circuit 18, an aluminum block 19, input and output transducers20 and 21, respectively, and an electronic feedback circuit 22.

Summing circuit 18 includes a differential amplifier 23 having anoninverting input connected via a resistor 24 from source 17, aninverting input connected from a junction 25, and a feedback resistor 26connected from junction 25 to an output junction 27.

A switch 28 is provided having a grounded contact 29, a contact 30connected from the output of feedback circuit 22, and a pole 31. Aresistor 32 is connected from junction 25 to pole 31.

Junction 27 is connected to input transducer 20. Output transducer 21 isconnected to an inverting input of a differential amplifier 33 via aresistor 34 and a junction 35 in feedback circuit 22.

Another differential amplifier is shown at 36. Each of the amplifiers 33and 36 have their noninverting inputs grounded via resistors 45 and 46,respectively.

Amplifier 33 has an output junction 37. Amplifier 36 has input andoutput junctions 38 and 39, respectively.

Amplifier 33 has feedback including a resistor 40 connected in parallelwith a capacitor 41 and between junctions 35 and 37.

A resistor 42 is connected between junctions 37 and 38. A feedbackresistor 43 is connected between junctions 38 and 39. Junction 39 isconnected to switch contact 30.

A portion of the apparatus of FIG. 2 is employed in the verification ofcircuit values used in electronic feedback circuit 22.

Switch pole 31 is maintained in the position shown and V_(out) (t) ismeasured on output junction 44 to obtain data to plot a Laplace transferfunction G(ω).

Switch pole 31 is then moved into engagement with contact 30 forvibration damping operation.

Circuit values for FIG. 2 arrived at according to the present inventionfollow:

Capacitor 41: 100 pfds.

Resistor 24: 15,000 Ohms

Resistor 26: 470,000 Ohms

Resistor 32: 15,000 Ohms

Resistor 34: 150,000 Ohms

Resistor 40: 47,000 Ohms

Resistor 42: 15,000 Ohms

Resistor 43: 150,000 Ohms

Resistor 45: 47,000 Ohms

Resistor 46: 15,000 Ohms

DAMPING CONDITIONS

In accordance with the present invention, devices may be constructedwhich demonstrate wide bandwidth active damping (WBAD) of a lightlydamped, multiple mode mechanical structure.

The technique relies on the method of coupling a mechanical structurewith multiple resonant modes to a damped electrical filter and amplifierso as to extract power from the mechanical structure and dissipate thepower in the electrical circuit.

This technique is unique and further different from prior techniques.

The mode sensing transducers are used to sense the displacements of theresonant modes of the structure, rather than rigid body displacements ofthe structure and rather than velocity or acceleration components of thestructure. Mode sensing transducers include electromechanicaltransducers, such as piezoelectric transducers, as well as acoustictransducers, such as microphones. Mode driving transducers includepiezoelectric transducers, as well as electric motors.

A positive feedback circuit with Laplace transfer function H(ω)including transducers 20 and 21, that generates the output signal thatdamps the resonant modes of the structure, operates primarily with phaselag between zero degrees to +180 degrees over the frequency range of themodes to be damped (where 90 degrees phase lag is preferred).

Conditions imposed on the electronic feedback circuit H(ω), obtainedfrom the Laplace transform by substitution of jω for Laplace variable sare shown below to be:

    G(ω)·H(ω)<1 for all ω>0 when Im[G(ω)·H(ω)]=0                      (1)

where the modes of G(ω) are described below, and

    c.sub.o Re[H(ω)]<1. for all ω0 when Im[H(ω)]=0 (2)

where Im(x) and Re(y) denote the real components of complex variables xand y, respectively, c_(o) is a constant,

    c.sub.n Im[H(ω)]<0. for all ω>0, n=1 . . .     (3)

and n refers to the mode of resonant frequency n which is to be activelydamped. The description in the body hereof is restricted to the case forn=1.

Note that if the Laplace transform of circuit 22 (FIG. 2) is G_(s) (s),then G(ω)=G_(s) (jω)

Similarly,

    H(ω)=H.sub.s (jω)

The constant c_(o) is defined in the Laplace transfer function G(ω)thus: ##EQU2## where the Greek letter sigma conventionally means "thesum of," the "sum" is the sum of the fractions produced by substitutingn=1, n=2, n=3, and so forth until n=K. K is the number of modal resonantfrequencies being controlled, and θ is the Greek letter zeta with thesubscript n.

It is conventional and well known to use the Greek sigma and the symbolsof equation (4) above and below the Greek sigma. It is also conventionalto use this with a function to the right thereof having constants withsubscripts n to identify different corresponding values of the constantsfor different values of n.

It is old and well known that each of the values of n are positiveintegers.

The value of K is the highest value of n. This is old and well known.The value of K is also a positive integer because n is always a positiveinteger and K is the maximum value of n. This is also old and wellknown.

The value of K is defined as the number of modes controlled. A mode isdefined in many prior art references, including but not limited to the"American Institute of Physics" (Third Edition, 1972). This referencereads "A normal mode of vibration is a mode of an umdamped system. Ingeneral, any composite motion of the system is analyzable into asummation of its normal modes." An analysis of this summation of modesis described in the paper "Electrical Damping of Orthogonal BendingModes in a Cylindrical Mast" by Charles J. Swigert and Robert L.Forward, Journal of Spacecraft and Rockets, Vol. 18, No. 1, Jan-Feb.,1981, pp. 5-10, a companion paper to that cited in the prior artstatement set forth herein.

Parameter K is the number of modes, n=1, n=2, . . . , n=K, havingresonant frequencies ω₁, ω₂, . . . , ω_(K), respectively, that candescribe the composite motion of the structure over the range ofresonant frequencies ω_(n) that lie within the passband of theelectronic feedback circuit.

Actually, K may be very large. The feedback circuit H(s) can provideimproved stability of the structure G(s) over the passband of thefeedback circuit. Stability is provided by satisfying the threeconditions set forth herein for each mode with resonant frequency ω_(n).

The Greek zeta in equation (4) is old and well known in the prior art.It is called "relative damping." The identical Greek letter zeta is usedin the prior art as a symbol for relative damping.

In Van Nostrand's Scientific Encyclopedia (Sixth Edition), "relativedamping" is defined thus: "For an underdamped system, a numberexpressing the quotient of the actual damping of a second-order linearsystem or element and its critical damping . . . . When the timeresponse to an abrupt stimulus is as fast as possible without overshoot,the response is said to be critically damped

underdamped when overshoot occurs

overdamed when response is slower than critical."

The technique can operate to damp the modes over either a wide bandwidthor a narrow bandwidth. Operating over a wide bandwidth, the WBAD candamp many modes in a structure (though not necessarily all of them) overa wide band of mechanical resonances, e.g., over more than an octave ofmodal resonant frequencies of the structure. Narrowing the bandwidth ofthe damper to a selected band of resonant modes, the mechanical powerbeing extracted from the modes can be absorbed by an electrical filterwith a lower power rating than would be required if all the modes wereto be similarly damped simultaneously.

Damping the n^(th) mode, for n=1, 2, . . . generally requires thecondition on c_(o) and c_(n) that

    sgn(c.sub.o)=-sgn(c.sub.n)

This condition can be obtained by placing two strain transducersadjacent to each other with polarities matched on a conducting surface.Both electrodes at one end are grounded together. The electrodes on theother end of the two transducers are used to inject the input voltageV_(in) (t) and to receive the output voltage V_(out) (t).

The wide bandwidth active damper (WBAD) of the present invention can beused to damp the vibrations on:

(a) large, flat rigid surfaces, such as windows;

(b) column or beam-like structures, such as water pipes or I-beams;

(c) chassis and shaker components of structures being shaken orvibrated, like printers, disk drives, vehicles, telescopes and radarantennae.

An aluminum block 19 (FIG. 2) (4"×1.5"×1") with piezoceramic straingauges 20 and 21 (Piezo Electric Products P/N SG-4M) are epoxy bonded tothe block as shown in FIG. 2. The block 19 is suspended on rubber bandsto minimize modal damping. Each of the strain gauge transducers 20 and21 produce an output voltage when strained by the block, and eachtransducer will apply a stress to the block when a voltage is applied tothe transducer. The piezoelectric transducers 20 and 21 are used foractive damping of the block's modes of vibration.

Modal resonant frequencies of the block 19 may include:

f₁ =24,770 Hz

f₂ =54,300 Hz

f₃ =71,800 Hz

f₄ =108.7 kHz

f₅ =123.0 kHz

f₆ =151.5 kHz

f₇ =185.0 kHz

f₈ =218.0 kHz

Note: ω_(n) =2πf_(n)

The electronic circuit 22 consists of two National LF356N operationalamplifiers 33 and 36, assorted resistors and capacitors and a HeathkitSine-Square Audio Generator (Model IG-18) to generate V_(in) (t). Theelectronic circuit is shown in an open loop configuration, since theswitch 28 is in the OFF position. This circuit, with the switch 28 inthe OFF position, allows measurement of G(s) for selected modes. Table Ilists ##EQU3## for the resonant mode f=24,770 Hz. Voltage amplitude andphase was measured on a Tektronix Model 531A oscilloscope.

                  TABLE I                                                         ______________________________________                                        TRANSFER FUNCTION OF STRUCTURE AND TRANS-                                     DUCERS FOR RESONANT MODE AT 24,770 Hz.                                                  ##STR1##                                                                       Real (in phase                                                                            Imaginary (out of                                      f          magnitude)  phase magnitude)                                       ______________________________________                                        24,300     .0071       .0000                                                  24,500     .0071       .0000                                                  24,620     .0064       .0000                                                  24,650     .0057       .0000                                                  24,735     .0013       .0013                                                  24,755     -.0063      .0063                                                  24,758     -.0076      .0076                                                  24,770     .0083       .031                                                   24,774     .023        .023                                                   24,780     .019        .009                                                   24,785     .018        .007                                                   24,800     .0129       .0035                                                  24,850     .0098       .0008                                                  25,000     .0089       .0000                                                  ______________________________________                                    

Measurement of the data in Table I is made for parameter estimation ofc_(o), c₁, ω₁ and θ₁ in G₁ (s) transfer function ##EQU4## Replacement ofs with jω in eqn (5) gives eqn (6) ##EQU5## A good fit of the measuredG(ω) data in Table I with the model of G₁ (ω) in eqn (6) is shown inFIG. 3. Parameter estimates are:

    c.sub.o =0.0079                                            (7)

    ω.sub.1 =115.6×10.sup.3 r/s(f.sub.1 =24,770 Hz) (8)

    c.sub.1 /ω.sub.1.sup.2 =-18×10.sup.-6 (c.sub.1 =-436×10.sup.3)                                     (9)

    2θ.sub.1 =0.00058                                    (10)

Note from the analysis discussions

    sgn(c.sub.o)=-sgn(c.sub.1)                                 (11)

and that ##EQU6## These structural/transducer characteristics allowcircuit gain H_(o) to be large and not violate the stability limits.Parameter c_(o) can be changed by changing the relative location andorientation of the transducers. Parameter c₁ can be changed by shiftingthe transducers 20 and 21 on the structure to vary the coupling to eachother.

Thus, circle 47 in FIG. 3 is a good curve fit to points 51-58 of thedata.

THE FIG. 2 EMBODIMENT

A wide bandwidth electronic damping circuit transfer function H(ω) willbe constructed according to the following to damp the resonant modes atfrequencies f₁ =24,770 Hz and f₂ =54,300 Hz. Damping f₁ is based on themeasured parameters of G(f) at frequency f₁.

An evaluation of the stability of H(ω) is made experimentally to testwhether other modes may be unstable, i.e., and oscillate. The unstablemode will oscillate at its resonant frequency, which can then bemeasured. This resonant mode can then be experimentally characterizedand circuit H₁ (ω) modified to regain stability of this unstable mode.To gain stability of this resonant mode, it is necessary to modify H(ω)by changing its gain or pole and zero locations to satisfy conditions(1), (2) and (3) for stability of this new mode. As each unstable modeis stabilized, the next (less) unstable mode in sequence will oscillateand require stabilization. When all of the modes are stable, none ofthese modes will oscillate.

The circuit transfer function to be evaluated for wide bandwidth dampingis ##EQU7## Since the parameters estimated in equations (7) to (10)yield the conditions ##EQU8## then circuit gain H_(o) can be largewithout violating the stability limits.

Resonant mode f₁ is best damped with s_(o) ≈ω₁. However, good damping ofthe second resonant mode at f₂ =54,300 Hz is desired as well. Thus s_(o)is placed at 33 KHz, intermediate between f₁ and f₂. The value of s_(o)is

    s.sub.o =2π (33 kHz)=220×10.sup.3 r/s             (14)

This choice of s_(o) provides 45° of phase lag at 33 kHz with minimumreduction in gain at 33 kHz. Consequently, at f₁ =24,770 Hz, H₁ (f₁)provides 35° of phase lag at a loop gain of 1.4, as shown in Table II.Table II illustrates that active damping of resonant mode f₁ reduces themodal response by ×0.44. At f₂ =53,300 Hz, H₁ (f₂) provides 57° of phaselag at a loop gain of 11.0 (Table II). Active damping of resonant modef₂ reduces, i.e., scales, the modal response to a factor of ×0.082, or8.2 percent of the mode's undamped response.

FIG. 2 illustrates an electronic feedback circuit to damp the aluminumblock structure with ##EQU9## With the switch 28 set to OFF, thefeedback circuit is interrupted, and allows the undamped resonantresponse to the input excitation V_(in) (t) to be measured at V₃ (t).Turning the switch to ON, the feedback circuit is completed and allowsdamping of the resonant response to V_(in) (t). For the same inputexcitation V_(in) (t), closed loop positive feedback provides asignificant reduction in the modal response, as measured at V₃ (t).

                  TABLE II                                                        ______________________________________                                        OPEN AND CLOSED LOOP RESPONSE OF STRUCTURE                                    WITH WIDE BANDWIDTH ELECTRONICS (K = 3 MODES)                                         Open Loop   Closed Loop                                                       Response    Response    Modal Response                                 f[Hz]                                                                                 ##STR2##                                                                                  ##STR3##                                                                                  ##STR4##                                     ______________________________________                                        24,000  0.88        0.50        0.57                                          f.sub.1 = 24,770                                                                      1.42        0.62        0.44                                          25,000  0.89        0.54        0.61                                          f.sub.2 = 54,300                                                                      11.0        0.90        0.082                                         f.sub.3 = 71,800                                                                      0.80        0.65        0.81                                          ______________________________________                                    

Table II lists the amplitudes for the undamped response of the open loopcircuit G(ω).H(ω) with switch 28 OFF, and for the damped response of theclosed loop circuit G(ω).H(ω) with switch 28 ON. The fractionalreduction in modal response is given in the last column. It isinteresting to note that the mode with the largest open loop response,at the 54.3 kHz resonant frequency, is the most strongly damped. All ofthe other modes are significantly damped, e.g., by 20 percent or more.Further damping can be obtained by raising the gain, increasing the H(ω)phase lag, or adding additional active damping circuits.

A moderate bandwidth electronic damping (feedback) circuit transferfunction H₂ (ω) will now be constructed. The circuit H₂ (ω) will beemployed to damp the resonant modes at the frequencies f₁ =24,770 Hz andf₂ =54,300 Hz. The experimental design for damping f₁ is based on themeasured parameters of G(f) at f₁. The test for stability of H₂ (ω) ismade experimentally to determine what other modes are unstable andoscillate.

If desired, the system in FIG. 4 may be the same as that shown in FIG. 2except for an electronic feedback circuit 60. Even circuits 22 and 60are identical except for the addition of a 22 picofarad capacitor 61.

THE FIG. 4 EMBODIMENT

The circuit transfer function to be evaluated for moderate bandwidthdamping is ##EQU10##

To allow H_(o) to be large in magnitude, we need

    sgn(c.sub.o)=-sgn(c.sub.1)                                 (17)

and

    sgn(c.sub.1)=sgn(H.sub.o)                                  (18)

This provides

    sgn(c.sub.1 H.sub.o)>0                                     (19)

and

    sgn(c.sub.o H.sub.o)<0                                     (20)

For arbitrarily large H_(o), ##EQU11## These conditions are satisfied bythe parameters for the structural resonance at f₁ with the circuittransfer function H₂ (ω) provided by the circuit in FIG. 4. The circuithas transfer function ##EQU12##

The resonant mode f₁ is best damped with w_(o) ≈1.7ω₁ when θ_(o) =1.i.e., s_(o) =s₁ =ω_(o) with H₂ (s) critically damped. Good damping ofthe second resonant mode is desired at f₂ =54,300 Hz. With ω₁ =2π(24,770Hz)=1.56×10⁵ then ω_(o) ≅1.7ω₁ =2.7×10⁵ is wanted, for dampoing of f₁.By placing s_(o) and s₁ above and below this ω_(o) estimate, we broadenthe damping bandwidth. Thus, we place s_(o) =2π(33 kHz) near the firstresonance at f₁ =24.77 kHz and we place s₁ =2π(48.2 kHz) near the secondresonance at 54.3 kHz to obtain good damping of both modes.

Table III shows this choice of s_(o) and s₁ provides a loop gain of 2.4at resonance f₁ =24.77 kHz with 64° of phase lag. Closed loop dampingscales mode f₁ response by×0.35. Table III shows resonance f₂ =54.3 kHzhas an open loop gain of 8.0 with 106° of phase lag. Closed loop dampingscales the modal response by 0.088 to 8.8 percent of its undampedmagnitude.

The limited damping bandwidth of H₂ (ω) is illustrated in Table III bythe reduction in natural damping at the resonance f₃ =71.8 kHz where anincrease in the modal response is seen by×1.97. This occurs because acondition is being violated. However, the closed loop system remainsstable. In those applications where there is minimal excitation of thesystem at these high resonant frequencies, e.g., at f₃ =71.8 kHz, it isuseful to limit the bandwidth over which the damper circuit needs tooperate. The two pole circuit H₂ (ω) provides this capability to limitthe damping action to the desired bandwidth of frequencies.

                  TABLE III                                                       ______________________________________                                        OPEN AND CLOSED LOOP RESPONSE OF STRUCTURE                                    WITH MODERATE BANDWIDTH ELECTRONICS                                           (N = 2 in FIG. 1)                                                                                           Model Response                                                                Reduction                                        f[Hz]                                                                                 ##STR5##                                                                                 ##STR6##                                                                                 ##STR7##                                       ______________________________________                                        24,000  1.35       0.82       0.61                                            24,770  2.40       0.85       0.35                                            54,300  8.00       0.70       0.088                                           71,800  0.62       1.22       1.97                                            ______________________________________                                    

PROOF

The plots of FIGS. 5-8 illustrate that the three conditions for dampingdescribed above are met in FIGS. 2 and 4. Equation (1) is:

    G(ω).H(ω)<1 for all ω>0 that Im[G(ω).H(ω)]=0 (1)

This is condition (1), or C₁.

Equation (2) is:

    c.sub.o Re[H(ω)]<1 for all ω>0 that Im[H(ω)]=0 (2)

This is condition (2), or C₂.

Equation (3) is:

    c.sub.n Im[H(ω)]<0 for all ω>0, for each of the modes, n=1, 2, . . .                                                     (3)

This is condition (3), C₃. This condition is specialized to a discussionfor the first mode, n=1, though it must be satisfied by all the modes.

For condition (3)(C₃), c₁ Im[H(ω)] is plotted in FIGS. 5 and 7 for thetwo circuits H₁ (s) and H₂ (s) described previously. These plots showthat conditions C₃ that c₁ Im[H(ω)]<0 is satisfied H₁ (ω) and H₂ (ω) forall ω>0.

For condition (2),(C₂), c₁ Im[H(ω)] as plotted is used to determine thatω when Im[H(ω)] is zero. This ω is used to evaluate c_(o) Re[H(ω)] andshow that c_(o) Re[H(ω)]<1 for those ω for which Im[H(ω)]=0. (No plotfor C₂).

For condition (1), (C₁), Im[G(ω).H(ω)] is plotted in FIGS. 6 and 8 tolocate where (which ω) that Im[G(ω).H(ω)]=0.

For those ω that satisfy the zero condition, an evaluation is made andit is shown that G(ω).H(ω)<1 for both H₁ (ω) and H₂ (ω).

Functions used are: ##EQU13##

H₁ (s) CONDITIONS

FIG. 5 illustrates condition C₃ is satisfied for H₁ (ω) for mode 1.

That is, ##EQU14##

For condition C₂, identify those ω for which Im[H₁ (ω)]=0. This occursat ω=0 and ω=∞. Evaluating c_(o) Re[H₁ (ω)], ##EQU15##

This is less than 1.0. ##EQU16##

This is less than 1.0.

Thus, condition C₂ is satisfied for H₁ (s).

FIG. 6 illustrates the plot Im[G(ω).H₁ (ω)]. While requiring a bit ofwork, it shows in FIG. 6 that

    P.sub.1 (ω)=Im[G(ω).H.sub.1 (ω)]=0

at ω_(z) =0, ∞, 1.553822×10⁵ r/s, 1.556403×10⁵ r/s (f=24,730 Hz)(f=24,771 Hz)

Evaluation of P_(o) (ω_(z))=Re[G(ω_(z)).H₁ (ω_(z))] for each of theseω_(z) =G(ω_(z)).H₁ (ω_(z)) gives

    P.sub.o (ω.sub.z =0)=-0.788

    P.sub.o (ω.sub.z =1.553822×10.sup.5)=-0.173

    P.sub.o (ω.sub.z =1.556403×10.sup.5)=-2.33

    P.sub.o (ω.sub.z =∞)=0.0

Thus, all P_(o) (ω_(z))<1, satisfying C₁.

Thus, condition C₁ is satisfied for H₁ (s), thus satisfying all of thedesign conditions (at least for the first mode, by C₃). Damping andstability of the other modes require verification of C₃ and C₁ for n=2,. . . and can be done experimentally.

H₂ (s) CONDITIONS

FIG. 7 illustrates condition C₃ is satisfied for H₂ (ω) for mode 1. Thatis, ##EQU17## for all ω>0.

For condition C₂, identify those ω for which Im[H₂ (ω)]=0. This occursat ω=0 and ω=∞. Evaluating c_(o) Re[H₁ (ω)],

    P.sub.2 (ω=0)=c.sub.o Re[H.sub.2 (ω=0)]=-1.69

This is less than 1.0 ##EQU18## This is less than 1.0.

Thus, condition C₂ is satisfied for H₂ (s), as is C₁ above.

FIG. 8 illustrates the plot of P₁ (ω)=Im[G(ω).H₂ (ω)]. While requiringeven more work, it shows in FIG. 8 that

    P.sub.1 (ω)=Im[G(ω).H.sub.2 (ω)]=0

at ω_(z) =0, ∞, 1.554125×10⁵, 1.556100×10⁵ (f=24,735 Hz) (f=24,766 Hz)

Evaluation of P_(o) (ω_(z))=G(ω_(z)).H₂ (ω_(z)) for each of these ω_(z)=Re[G(ω_(z)).H₂ (ω_(z))] gives

    P.sub.o (ω.sub.z =0)=-1.69

    P.sub.o (ω.sub.z =1.554125×10.sup.5 r/s)=-0.289

    P.sub.o (ω.sub.z =1.556100×10.sup.5 r/s)=-5.03

    P.sub.o (ω.sub.z =∞)=0.0

Thus, all these P_(o) (ω_(z))<1 so that condition C₁ is satisfied for H₂(s). Thus, all the design conditions are satisfied for damping andstability of the first mode.

What is claimed is:
 1. A system for wide bandwidth damping, said systemcomprising:a vibratable mechanical structure to be damped; an inputtransducer responsive to an input signal to affect vibration of saidmechanical structure; an output transducer responsive to displacementvibration of said mechanical structure for producing an output signal inaccordance therewith; an active electronic feedback circuit connectedfrom said output transducer for producing a positive feedback signal toeach resonant mode to be damped; a signal souce; and an input circuitfor impressing an input signal on said input transducer directlyproportional to the sum of the output signals of said source and saidfeedback circuit, said feedback circuit supplying said feedback signalto said input circuit with a phase lagging the output signal of saidoutput transducer by more than 0° and less than 180°, wherein: saidfeedback signal has a phase which lags said output transducer outputsignal by approximately 90°, and wherein: said input and outputtransducers are vibration sensitive transducers fixed to said structurein spaced relative positions; said electronic feedback circuit isconnected from said output transducer to said input transducer, saidcircuit having a Laplace transfer function H(ω) where ω is radianfrequency, said circuit having circuit components with values such thatH(ω) meets conditions (1), (2) and (3) as follows:

    G(ω)H(ω)<1 for all ω when Im[G(ω)H(ω)]=0 (1)

G(ω) being the Laplace transfer function through said input transducer,through said structure and through said output transducer, Im[G(ω)H(ω)]being the imaginary part of the complex product G(ω)H(ω), (2) c_(o)Re[H(ω)]<1, for all ω>0 that Im[H(ω)]=0, c_(o) being a constant,Re[H(ω)] being the real part of the complex variable H(ω), and Im[H(ω)]being the imaginary part of H(ω), and (3) c_(n) Im[H(ω)]>0 for all ω>0and one or more of the modes to be damped, where for each said mode n=1,2, . . . , each c_(n) being a constant, Im[H(ω)] being the imaginarypart of the complex variable H(ω).
 2. A system for wide bandwidthdamping as claimed in claim 1,c_(o) being defined in G(ω) as follows:##EQU19## where K is the number of resonant modes controlled, and θ isrelative damping.
 3. A system for wide bandwidth damping, said systemcomprising:a vibratable mechanical structure to be damped; an inputtransducer responsive to an input signal to affect vibration of saidmechanical structure; an output transducer responsive to displacementvibration of said mechanical structure for producing an output signal inaccordance therewith; an active electronic feedback circuit connectedfrom said output transducer for producing a positive feedback signal toeach resonant mode to be damped; a signal source; and an input circuitfor impressing an input signal on said input transducer directlyproportional to the sum of the output signals of said source and saidfeedback circuit, said feedback circuit supplying said feedback signalto said input circuit with a phase lagging the output signal of saidoutput transducer by more than 0° and less than 180°, wherein: saidinput and output transducers are vibration sensitive transducers fixedto said structure in spaced relative positions; said electronic feedbackcircuit is connected from said output transducer to said inputtransducer,said circuit having a Laplace transfer function H(ω) where ωis radian frequency, said circuit having circuit components with valuessuch that H(ω) meets conditions (1), (2) and (3) as follows:

    G(ω)H(ω)<1 for all ω when Im[G(ω)H(ω)]=0 (1)

G(ω) being the Laplace transfer function through said input transducer,through said structure, and through said output transducer, Im[G(ω)H(ω)]being the imaginary part of the complex product G(ω)H(ω), (2) c_(o)Re[H(ω)]<1, for all ω>0 that Im[H(ω)]=0, c_(o) being a constant,Re[H(ω)] being the real part of the complex variable H(ω), and (3) c_(n)Im[H(ω)]<0 for all ω>0 and one or more of the modes to be damped, wherefor each said mode n=1, 2, . . . , each c_(n) being a constant, Im[H(ω)]being the imaginary part of the complex variable H(ω).
 4. A system forwideband damping as claimed in claim 3, wherein:c_(o) is defined in G(ω)as follows: ##EQU20## where K is the number of resonant modescontrolled, and θ is relative damping.
 5. In a system for dampingmechanicl vibrations, the combination comprising:a vibratable mechanicalstructure; input and output vibration sensitive transducers fixed tosaid structure in spaced relative positions; an electronic feedbackcircuit connected from said output transducer to said input transducer,said circuit having a Laplace transfer function H(ω) where ω is radianfrequency, said circuit having circuit components with values such thatH(ω) meets conditions (1), (2) and (3) as follows:

    G(ω)H(ω)<1 for all ω when Im[G(ω)H(ω)]=0 (1)

G(ω) being the Laplace transfer function through said input transducer,through said structure and through said output transducer, Im[G(ω)H(ω)],being the imaginary part of the complex product G(ω)H(ω), (2) c_(o)Re[H(ω)]<1, for all ω>0 that Im[H(ω)]=0, c_(o) being a constant,Re[H(ω)] being the real part of the complex variable H(ω), and (3) c_(n)Im[H(ω)]<0 for all ω>0, and one or more of the modes to be damped, wherefor each said mode n=1, 2, . . . , each c_(n) being a constant, Im[H(ω)]being the imaginary part of the complex variable H(ω).
 6. In a systemfor damping mechanical vibrations as claimed in claim 5, wherein:c_(o)is defined in G(ω) as follows: ##EQU21## where K is the number ofresonant modes controlled, and θ is relative damping.
 7. The method ofdamping mechanical vibrations, said method comprising the stepsof:providing a vibratable structure; mounting input and output vibrationtransducers on said structure; empirically determining the constantsc_(o) and c₁ of the transfer function G(ω) through said inputtransducer, through said structure and through said output transducer,where: ##EQU22## where K is the number of resonant modes controlled, andθ is the relative damping; and connecting an electronic feedback circuitfrom said output transducer to said input transducer, said circuithaving a Laplace transfer function H(ω) where ω is radian frequency,said circuit having circuit components with values such that H(ω) meetsconditions (1), (2) and (3), as follows:

    G(ω)H(ω)<1 for all ω when Im[G(ω)H(ω)]=0 (1)

Im[G(ω)H(ω)] being the imaginary part of the complex product G(ω)H(ω),(2) c_(o) Re[H(ω)]<1, for all ω>0 that Im[H(ω)]=0, Re[H(ω)] being thereal part of the complex variable H(ω), and (3) c_(n) Im[H(ω)]<0 for allω>0, and one or more of the modes to be damped, where for each said moden=1, 2, . . . , Im[H(ω)] being the imaginary part of the complexvariable H(ω).
 8. The method of damping mechanical vibrations as claimedin claim 7, wherein:said feedback circuit supplies a feedback signal tosaid input transducer that has a phase lagging the output signal of saidoutput transducer more than 0° and less than 180°.
 9. The method ofdamping mechanical vibrations as claimed in claim 8, wherein:saidfeedback signal has a phase that lags the output signal of said outputtransducer approximately 90°.